Potential automorphy and change of weight

Abstract

We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call 'potential diagonalizability'. This result allows for 'change of weight' and seems to be substantially more flexible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real field is potentially automorphic, and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

Crystalline liftings and the weight part of Serre’s conjecture

“…Also, let W diag (ρ) be the set of Serre weights a, such that ρ has a potentially diagonalizable crystalline lift of Hodge type a. Here, potential diagonalizability is in the sense of [BLGGT14], and the precise definition can be found in Subsection 1.4 of loc. cit., which we omit.…”

“…Finally in Section 8, we apply our local results to weight part of Serre's conjecture. The application is straightforward, using automorphy lifting theorems proved by [BLGGT14].…”

On critical values of L-functions of potentially automorphic motives

“…This is expected to hold in our setting, and many, if not most, cases have been already proved [22][23][24]. Combined with Hypothesis (2), this implies that we can express the L-function as that of a cohomological automorphic representation on a totally definite unitary group. While doing the motivic computations behind the expression of c + (M(χ )(k)), we draw some consequences of the general formula proved in [14] for the case of arbitrary critical intervals.…”

“…Moreover, if w 0 ∈ Z satisfies that w 0 ≡ a τ (2) for one (or every) τ ∈ J L , then χ can be taken to have weight w 0 .…”

“…Let be a CM type for L/K , that is, ⊂ J L consists of a choice of one of the two possible extensions of each σ ∈ J K to L. Proposition 2.1.1. Let (a τ ) τ ∈ be a tuple of integers such that a τ ≡ a τ (2) for every τ, τ ∈ . Then there exists an algebraic Hecke character χ of L of infinity type (n τ ) τ ∈J L such that…”